15 Equation 906

In this chapter we study finite magmas that obey equation 906,

\begin{equation} \label{906} x = y \diamond ((y \diamond x) \diamond (x \diamond x)) \end{equation}

for all \(x,y\). We can write this as

\begin{equation} \label{906-a} L_y (L_y x \diamond Sx) = x. \end{equation}

This implies that \(L_y\) is surjective, hence invertible by finiteness, so

\begin{equation} \label{906-b} L_y x \diamond Sx = L_y^{-1} x. \end{equation}

Corollary 15.1 Edge disjointness of left cycles

For any integer \(n\),

\[ L_y x = L_z x \implies L_y^{n} x = L_z^{n} x. \]

Proof ▶

This is trivial for \(n=0,1\), and \(n=-1\) follows from Equation 3. Observe that if the claim holds for \(n=0,-1,\dots ,-m\) for any \(m \geq 1\) then it also holds for \(n=-m-1\). Finally, since there is a common period to all the \(L_y\) by finiteness (or Legendre’s theorem), the set of \(n\) for which the claim holds is periodic. The claim follows.

Setting \(n = N-2\) for \(N{\gt}2\) a common period of \(L_y,L_z\) (which exists by finiteness) we conclude that

\begin{equation} \label{lyzx} L_y x = L_z x \implies L_y^2 x = L_z^2 x. \end{equation}

Theorem 15.2

✓

For finite magmas, equation 906 implies equation 3862,

\begin{equation} \label{3862} (x \diamond (x \diamond x)) \diamond x = x \diamond x. \end{equation}

Proof ▶

Observe from Equation 2 that

\[ L_x S^2 x = L_x (L_x x \diamond Sx) = x \]

while from Equation 3 we have

\[ L_{L_x Sx} S^2 x = L_x Sx \diamond S S x = L_x^{-1} Sx = x. \]

Thus

\[ L_x S^2 x = L_{L_x Sx} S^2 x = x \]

and hence by the \(n=2\) case of Corollary 15.1

\[ L_x x = L_{L_x Sx} x \]

giving the claim.